LMFDB - Embedded newform 72.7.b.c.19.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [72,7,Mod(19,72)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(72, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 1, 0]))

N = Newforms(chi, 7, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("72.19");

S:= CuspForms(chi, 7);

N := Newforms(S);

Level:	$ N $	$=$	$ 72 = 2^{3} \cdot 3^{2} $
Weight:	$ k $	$=$	$ 7 $
Character orbit:	$[\chi]$	$=$	72.b (of order $2$, degree $1$, minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$16.5638940206$
Analytic rank:	$0$
Dimension:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - 2 x^{11} + 31 x^{10} - 1286 x^{9} + 7702 x^{8} - 174032 x^{7} + 1952056 x^{6} + \cdots + 767595744 $ x^12 - 2x^11 + 31x^10 - 1286x^9 + 7702x^8 - 174032x^7 + 1952056x^6 - 6345392x^5 + 7695616x^4 - 6850848x^3 - 19274256x^2 - 69390432*x + 767595744
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{30}\cdot 3^{11} $
Twist minimal:	no (minimal twist has level 24)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			19.4
Root			$-1.87190 - 1.33932i$ of defining polynomial
Character	$\chi$	$=$	72.19
Dual form			72.7.b.c.19.3

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-7.11414 + 3.65910i) q^{2} +(37.2219 - 52.0627i) q^{4} +87.0704i q^{5} -355.505i q^{7} +(-74.2991 + 506.580i) q^{8} +O(q^{10})$	\(q+(-7.11414 + 3.65910i) q^{2} +(37.2219 - 52.0627i) q^{4} +87.0704i q^{5} -355.505i q^{7} +(-74.2991 + 506.580i) q^{8} +(-318.599 - 619.431i) q^{10} +185.285 q^{11} +2323.93i q^{13} +(1300.83 + 2529.11i) q^{14} +(-1325.06 - 3875.75i) q^{16} -7535.06 q^{17} -807.031 q^{19} +(4533.12 + 3240.93i) q^{20} +(-1318.14 + 677.978i) q^{22} -15162.9i q^{23} +8043.75 q^{25} +(-8503.52 - 16532.8i) q^{26} +(-18508.6 - 13232.6i) q^{28} -32494.6i q^{29} -44109.9i q^{31} +(23608.4 + 22724.1i) q^{32} +(53605.5 - 27571.6i) q^{34} +30954.0 q^{35} -59093.4i q^{37} +(5741.33 - 2953.01i) q^{38} +(-44108.1 - 6469.25i) q^{40} -87145.9 q^{41} -124698. q^{43} +(6896.68 - 9646.45i) q^{44} +(55482.5 + 107871. i) q^{46} +64962.9i q^{47} -8735.01 q^{49} +(-57224.4 + 29432.9i) q^{50} +(120990. + 86501.3i) q^{52} -154374. i q^{53} +16132.9i q^{55} +(180092. + 26413.7i) q^{56} +(118901. + 231171. i) q^{58} -41543.6 q^{59} +108362. i q^{61} +(161403. + 313804. i) q^{62} +(-251103. - 75277.0i) q^{64} -202346. q^{65} +482658. q^{67} +(-280469. + 392296. i) q^{68} +(-220211. + 113264. i) q^{70} -418654. i q^{71} -546155. q^{73} +(216229. + 420399. i) q^{74} +(-30039.2 + 42016.2i) q^{76} -65869.9i q^{77} +538230. i q^{79} +(337463. - 115373. i) q^{80} +(619968. - 318876. i) q^{82} -443538. q^{83} -656080. i q^{85} +(887119. - 456283. i) q^{86} +(-13766.5 + 93861.9i) q^{88} +151438. q^{89} +826171. q^{91} +(-789420. - 564391. i) q^{92} +(-237706. - 462155. i) q^{94} -70268.4i q^{95} -1.18214e6 q^{97} +(62142.0 - 31962.3i) q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q - 10 q^{2} + 24 q^{4} - 796 q^{8}+O(q^{10}) $ 12 * q - 10 * q^2 + 24 * q^4 - 796 * q^8	$ 12 q - 10 q^{2} + 24 q^{4} - 796 q^{8} + 2172 q^{10} - 2720 q^{11} + 6444 q^{14} + 11640 q^{16} + 4888 q^{17} + 3936 q^{19} + 31608 q^{20} - 60432 q^{22} - 27204 q^{25} - 53952 q^{26} - 57072 q^{28} - 109480 q^{32} + 47388 q^{34} - 162336 q^{35} + 89080 q^{38} + 72120 q^{40} + 54280 q^{41} - 49824 q^{43} - 229184 q^{44} + 171864 q^{46} - 304644 q^{49} + 500078 q^{50} + 256848 q^{52} + 699816 q^{56} - 409524 q^{58} + 886144 q^{59} - 691356 q^{62} - 500640 q^{64} - 473376 q^{65} + 1565952 q^{67} - 669104 q^{68} + 473784 q^{70} + 555480 q^{73} + 753720 q^{74} - 293136 q^{76} + 251616 q^{80} + 2317716 q^{82} - 2497760 q^{83} - 476024 q^{86} + 971424 q^{88} - 367400 q^{89} - 4475808 q^{91} + 377376 q^{92} - 2642568 q^{94} - 1165656 q^{97} - 182674 q^{98}+O(q^{100}) $ 12 * q - 10 * q^2 + 24 * q^4 - 796 * q^8 + 2172 * q^10 - 2720 * q^11 + 6444 * q^14 + 11640 * q^16 + 4888 * q^17 + 3936 * q^19 + 31608 * q^20 - 60432 * q^22 - 27204 * q^25 - 53952 * q^26 - 57072 * q^28 - 109480 * q^32 + 47388 * q^34 - 162336 * q^35 + 89080 * q^38 + 72120 * q^40 + 54280 * q^41 - 49824 * q^43 - 229184 * q^44 + 171864 * q^46 - 304644 * q^49 + 500078 * q^50 + 256848 * q^52 + 699816 * q^56 - 409524 * q^58 + 886144 * q^59 - 691356 * q^62 - 500640 * q^64 - 473376 * q^65 + 1565952 * q^67 - 669104 * q^68 + 473784 * q^70 + 555480 * q^73 + 753720 * q^74 - 293136 * q^76 + 251616 * q^80 + 2317716 * q^82 - 2497760 * q^83 - 476024 * q^86 + 971424 * q^88 - 367400 * q^89 - 4475808 * q^91 + 377376 * q^92 - 2642568 * q^94 - 1165656 * q^97 - 182674 * q^98

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/72\mathbb{Z}\right)^\times$.

$n$	$37$	$55$	$65$
$\chi(n)$	$-1$	$-1$	$1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$	−7.11414	+	3.65910i	−0.889267	+	0.457388i
$2$	−7.11414	+	3.65910i	−0.889267	+	0.457388i
$3$		0			0
$3$		0			0
$4$	37.2219	−	52.0627i	0.581593	−	0.813480i
$5$			87.0704i			0.696563i	0.937390	+	0.348281i	$0.113235\pi$
$5$			87.0704i			0.696563i	−0.937390	+	0.348281i	$0.886765\pi$
$6$		0			0
$7$		−	355.505i		−	1.03646i	−0.855242	−	0.518229i	$-0.826591\pi$
$7$		−	355.505i		−	1.03646i	0.855242	−	0.518229i	$-0.173409\pi$
$8$	−74.2991	+	506.580i	−0.145116	+	0.989415i
$9$		0			0
$10$	−318.599	−	619.431i	−0.318599	−	0.619431i
$11$	185.285			0.139208			0.0696038	−	0.997575i	$-0.477827\pi$
$11$	185.285			0.139208			0.0696038	+	0.997575i	$0.477827\pi$
$12$		0			0
$13$			2323.93i			1.05778i	0.848692	+	0.528888i	$0.177391\pi$
$13$			2323.93i			1.05778i	−0.848692	+	0.528888i	$0.822609\pi$
$14$	1300.83	+	2529.11i	0.474064	+	0.921689i
$15$		0			0
$16$	−1325.06	−	3875.75i	−0.323500	−	0.946228i
$17$	−7535.06			−1.53370			−0.766849	−	0.641827i	$-0.778177\pi$
$17$	−7535.06			−1.53370			−0.766849	+	0.641827i	$0.778177\pi$
$18$		0			0
$19$	−807.031			−0.117660			−0.0588301	−	0.998268i	$-0.518737\pi$
$19$	−807.031			−0.117660			−0.0588301	+	0.998268i	$0.518737\pi$
$20$	4533.12	+	3240.93i	0.566640	+	0.405116i
$21$		0			0
$22$	−1318.14	+	677.978i	−0.123793	+	0.0636718i
$23$		−	15162.9i		−	1.24623i	−0.782131	−	0.623115i	$-0.785867\pi$
$23$		−	15162.9i		−	1.24623i	0.782131	−	0.623115i	$-0.214133\pi$
$24$		0			0
$25$	8043.75			0.514800
$26$	−8503.52	−	16532.8i	−0.483814	−	0.940646i
$27$		0			0
$28$	−18508.6	−	13232.6i	−0.843138	−	0.602797i
$29$		−	32494.6i		−	1.33235i	−0.745797	−	0.666173i	$-0.767931\pi$
$29$		−	32494.6i		−	1.33235i	0.745797	−	0.666173i	$-0.232069\pi$
$30$		0			0
$31$		−	44109.9i		−	1.48065i	−0.672251	−	0.740323i	$-0.734673\pi$
$31$		−	44109.9i		−	1.48065i	0.672251	−	0.740323i	$-0.265327\pi$
$32$	23608.4	+	22724.1i	0.720471	+	0.693485i
$33$		0			0
$34$	53605.5	−	27571.6i	1.36387	−	0.701495i
$35$	30954.0			0.721958
$36$		0			0
$37$		−	59093.4i		−	1.16663i	−0.812245	−	0.583316i	$-0.801755\pi$
$37$		−	59093.4i		−	1.16663i	0.812245	−	0.583316i	$-0.198245\pi$
$38$	5741.33	−	2953.01i	0.104631	−	0.0538163i
$39$		0			0
$40$	−44108.1	−	6469.25i	−0.689190	−	0.101082i
$41$	−87145.9			−1.26443			−0.632216	−	0.774792i	$-0.717854\pi$
$41$	−87145.9			−1.26443			−0.632216	+	0.774792i	$0.717854\pi$
$42$		0			0
$43$	−124698.			−1.56839			−0.784195	−	0.620514i	$-0.786924\pi$
$43$	−124698.			−1.56839			−0.784195	+	0.620514i	$0.786924\pi$
$44$	6896.68	−	9646.45i	0.0809621	−	0.113243i
$45$		0			0
$46$	55482.5	+	107871.i	0.570010	+	1.10823i
$47$			64962.9i			0.625708i	0.949801	+	0.312854i	$0.101285\pi$
$47$			64962.9i			0.625708i	−0.949801	+	0.312854i	$0.898715\pi$
$48$		0			0
$49$	−8735.01			−0.0742463
$50$	−57224.4	+	29432.9i	−0.457795	+	0.235463i
$51$		0			0
$52$	120990.	+	86501.3i	0.860480	+	0.615195i
$53$		−	154374.i		−	1.03692i	−0.855101	−	0.518462i	$-0.826505\pi$
$53$		−	154374.i		−	1.03692i	0.855101	−	0.518462i	$-0.173495\pi$
$54$		0			0
$55$			16132.9i			0.0969668i
$56$	180092.	+	26413.7i	1.02549	+	0.150406i
$57$		0			0
$58$	118901.	+	231171.i	0.609399	+	1.18481i
$59$	−41543.6			−0.202278			−0.101139	−	0.994872i	$-0.532249\pi$
$59$	−41543.6			−0.202278			−0.101139	+	0.994872i	$0.532249\pi$
$60$		0			0
$61$			108362.i			0.477407i	0.971092	+	0.238704i	$0.0767224\pi$
$61$			108362.i			0.477407i	−0.971092	+	0.238704i	$0.923278\pi$
$62$	161403.	+	313804.i	0.677230	+	1.31669i
$63$		0			0
$64$	−251103.	−	75277.0i	−0.957883	−	0.287159i
$65$	−202346.			−0.736808
$66$		0			0
$67$	482658.			1.60478			0.802390	−	0.596801i	$-0.203562\pi$
$67$	482658.			1.60478			0.802390	+	0.596801i	$0.203562\pi$
$68$	−280469.	+	392296.i	−0.891988	+	1.24763i
$69$		0			0
$70$	−220211.	+	113264.i	−0.642014	+	0.330215i
$71$		−	418654.i		−	1.16971i	−0.811136	−	0.584857i	$-0.801151\pi$
$71$		−	418654.i		−	1.16971i	0.811136	−	0.584857i	$-0.198849\pi$
$72$		0			0
$73$	−546155.			−1.40393			−0.701967	−	0.712209i	$-0.747695\pi$
$73$	−546155.			−1.40393			−0.701967	+	0.712209i	$0.747695\pi$
$74$	216229.	+	420399.i	0.533603	+	1.03745i
$75$		0			0
$76$	−30039.2	+	42016.2i	−0.0684303	+	0.0957142i
$77$		−	65869.9i		−	0.144283i
$78$		0			0
$79$			538230.i			1.09166i	0.837896	+	0.545829i	$0.183785\pi$
$79$			538230.i			1.09166i	−0.837896	+	0.545829i	$0.816215\pi$
$80$	337463.	−	115373.i	0.659107	−	0.225338i
$81$		0			0
$82$	619968.	−	318876.i	1.12442	−	0.578336i
$83$	−443538.			−0.775705			−0.387852	−	0.921722i	$-0.626783\pi$
$83$	−443538.			−0.775705			−0.387852	+	0.921722i	$0.626783\pi$
$84$		0			0
$85$		−	656080.i		−	1.06832i
$86$	887119.	−	456283.i	1.39472	−	0.717363i
$87$		0			0
$88$	−13766.5	+	93861.9i	−0.0202012	+	0.137734i
$89$	151438.			0.214815			0.107408	−	0.994215i	$-0.465745\pi$
$89$	151438.			0.214815			0.107408	+	0.994215i	$0.465745\pi$
$90$		0			0
$91$	826171.			1.09634
$92$	−789420.	−	564391.i	−1.01378	−	0.724798i
$93$		0			0
$94$	−237706.	−	462155.i	−0.286191	−	0.556422i
$95$		−	70268.4i		−	0.0819577i
$96$		0			0
$97$	−1.18214e6			−1.29525			−0.647626	−	0.761958i	$-0.724238\pi$
$97$	−1.18214e6			−1.29525			−0.647626	+	0.761958i	$0.724238\pi$
$98$	62142.0	−	31962.3i	0.0660248	−	0.0339594i
$99$		0			0
$100$	299404.	−	418780.i	0.299404	−	0.418780i
$101$		−	74134.7i		−	0.0719544i	−0.999353	−	0.0359772i	$-0.988546\pi$
$101$		−	74134.7i		−	0.0719544i	0.999353	−	0.0359772i	$-0.0114544\pi$
$102$		0			0
$103$			246910.i			0.225957i	0.993597	+	0.112979i	$0.0360392\pi$
$103$			246910.i			0.225957i	−0.993597	+	0.112979i	$0.963961\pi$
$104$	−1.17726e6	−	172666.i	−1.04658	−	0.153500i
$105$		0			0
$106$	564870.	+	1.09824e6i	0.474276	+	0.922102i
$107$	1.51454e6			1.23632			0.618158	−	0.786054i	$-0.287879\pi$
$107$	1.51454e6			1.23632			0.618158	+	0.786054i	$0.287879\pi$
$108$		0			0
$109$		−	1.36271e6i		−	1.05226i	−0.850404	−	0.526130i	$-0.823643\pi$
$109$		−	1.36271e6i		−	1.05226i	0.850404	−	0.526130i	$-0.176357\pi$
$110$	−59031.8	−	114771.i	−0.0443514	−	0.0862294i
$111$		0			0
$112$	−1.37785e6	+	471064.i	−0.980726	+	0.335294i
$113$	−141201.			−0.0978596			−0.0489298	−	0.998802i	$-0.515581\pi$
$113$	−141201.			−0.0978596			−0.0489298	+	0.998802i	$0.515581\pi$
$114$		0			0
$115$	1.32024e6			0.868077
$116$	−1.69176e6	−	1.20951e6i	−1.08384	−	0.774883i
$117$		0			0
$118$	295547.	−	152012.i	0.179879	−	0.0925193i
$119$			2.67875e6i			1.58961i
$120$		0			0
$121$	−1.73723e6			−0.980621
$122$	−396509.	−	770905.i	−0.218360	−	0.424543i
$123$		0			0
$124$	−2.29648e6	−	1.64186e6i	−1.20448	−	0.861133i
$125$			2.06085e6i			1.05515i
$126$		0			0
$127$		−	880043.i		−	0.429628i	−0.976655	−	0.214814i	$-0.931085\pi$
$127$		−	880043.i		−	0.429628i	0.976655	−	0.214814i	$-0.0689145\pi$
$128$	2.06183e6	−	383282.i	0.983157	−	0.182763i
$129$		0			0
$130$	1.43952e6	−	740404.i	0.655219	−	0.337007i
$131$	1.45003e6			0.645005			0.322503	−	0.946569i	$-0.395476\pi$
$131$	1.45003e6			0.645005			0.322503	+	0.946569i	$0.395476\pi$
$132$		0			0
$133$			286904.i			0.121950i
$134$	−3.43370e6	+	1.76610e6i	−1.42708	+	0.734006i
$135$		0			0
$136$	559848.	−	3.81711e6i	0.222563	−	1.51746i
$137$	−427562.			−0.166279			−0.0831395	−	0.996538i	$-0.526495\pi$
$137$	−427562.			−0.166279			−0.0831395	+	0.996538i	$0.526495\pi$
$138$		0			0
$139$	1.61706e6			0.602119			0.301060	−	0.953605i	$-0.402660\pi$
$139$	1.61706e6			0.602119			0.301060	+	0.953605i	$0.402660\pi$
$140$	1.15217e6	−	1.61155e6i	0.419886	−	0.587299i
$141$		0			0
$142$	1.53190e6	+	2.97836e6i	0.535013	+	1.04019i
$143$			430591.i			0.147250i
$144$		0			0
$145$	2.82932e6			0.928063
$146$	3.88542e6	−	1.99844e6i	1.24847	−	0.642143i
$147$		0			0
$148$	−3.07656e6	−	2.19957e6i	−0.949032	−	0.678505i
$149$		−	5.96268e6i		−	1.80253i	−0.433268	−	0.901265i	$-0.642639\pi$
$149$		−	5.96268e6i		−	1.80253i	0.433268	−	0.901265i	$-0.357361\pi$
$150$		0			0
$151$			6.80101e6i			1.97534i	0.156537	+	0.987672i	$0.449967\pi$
$151$			6.80101e6i			1.97534i	−0.156537	+	0.987672i	$0.550033\pi$
$152$	59961.7	−	408826.i	0.0170743	−	0.116415i
$153$		0			0
$154$	241025.	+	468607.i	0.0659932	+	0.128306i
$155$	3.84067e6			1.03136
$156$		0			0
$157$			5.34814e6i			1.38199i	0.722861	+	0.690993i	$0.242827\pi$
$157$			5.34814e6i			1.38199i	−0.722861	+	0.690993i	$0.757173\pi$
$158$	−1.96944e6	−	3.82904e6i	−0.499311	−	0.970776i
$159$		0			0
$160$	−1.97860e6	+	2.05559e6i	−0.483056	+	0.501853i
$161$	−5.39048e6			−1.29166
$162$		0			0
$163$	−260355.			−0.0601178			−0.0300589	−	0.999548i	$-0.509569\pi$
$163$	−260355.			−0.0601178			−0.0300589	+	0.999548i	$0.509569\pi$
$164$	−3.24374e6	+	4.53705e6i	−0.735384	+	1.02859i
$165$		0			0
$166$	3.15539e6	−	1.62295e6i	0.689809	−	0.354798i
$167$			2.73821e6i			0.587920i	0.955818	+	0.293960i	$0.0949732\pi$
$167$			2.73821e6i			0.587920i	−0.955818	+	0.293960i	$0.905027\pi$
$168$		0			0
$169$	−573863.			−0.118891
$170$	2.40067e6	+	4.66745e6i	0.488635	+	0.950020i
$171$		0			0
$172$	−4.64150e6	+	6.49212e6i	−0.912165	+	1.27585i
$173$			5.87479e6i			1.13463i	0.823501	+	0.567315i	$0.192018\pi$
$173$			5.87479e6i			1.13463i	−0.823501	+	0.567315i	$0.807982\pi$
$174$		0			0
$175$		−	2.85960e6i		−	0.533569i
$176$	−245513.	−	718119.i	−0.0450336	−	0.131722i
$177$		0			0
$178$	−1.07735e6	+	554128.i	−0.191028	+	0.0982539i
$179$	1.20815e6			0.210650			0.105325	−	0.994438i	$-0.466412\pi$
$179$	1.20815e6			0.210650			0.105325	+	0.994438i	$0.466412\pi$
$180$		0			0
$181$		−	3.70353e6i		−	0.624568i	−0.949989	−	0.312284i	$-0.898906\pi$
$181$		−	3.70353e6i		−	0.624568i	0.949989	−	0.312284i	$-0.101094\pi$
$182$	−5.87750e6	+	3.02304e6i	−0.974941	+	0.501453i
$183$		0			0
$184$	7.68121e6	+	1.12659e6i	1.23304	+	0.180847i
$185$	5.14528e6			0.812632
$186$		0			0
$187$	−1.39614e6			−0.213502
$188$	3.38214e6	+	2.41804e6i	0.509001	+	0.363907i
$189$		0			0
$190$	257119.	+	499899.i	0.0374864	+	0.0728823i
$191$		−	1.42931e6i		−	0.205128i	−0.994726	−	0.102564i	$-0.967295\pi$
$191$		−	1.42931e6i		−	0.205128i	0.994726	−	0.102564i	$-0.0327047\pi$
$192$		0			0
$193$	−1.23148e6			−0.171299			−0.0856494	−	0.996325i	$-0.527296\pi$
$193$	−1.23148e6			−0.171299			−0.0856494	+	0.996325i	$0.527296\pi$
$194$	8.40992e6	−	4.32558e6i	1.15183	−	0.592433i
$195$		0			0
$196$	−325134.	+	454768.i	−0.0431811	+	0.0603979i
$197$			6.00987e6i			0.786079i	0.919522	+	0.393039i	$0.128576\pi$
$197$			6.00987e6i			0.786079i	−0.919522	+	0.393039i	$0.871424\pi$
$198$		0			0
$199$		−	3.62663e6i		−	0.460197i	−0.973167	−	0.230099i	$-0.926095\pi$
$199$		−	3.62663e6i		−	0.460197i	0.973167	−	0.230099i	$-0.0739048\pi$
$200$	−597644.	+	4.07481e6i	−0.0747055	+	0.509351i
$201$		0			0
$202$	271266.	+	527405.i	0.0329111	+	0.0639867i
$203$	−1.15520e7			−1.38092
$204$		0			0
$205$		−	7.58782e6i		−	0.880756i
$206$	−903468.	−	1.75655e6i	−0.103350	−	0.200936i
$207$		0			0
$208$	9.00699e6	−	3.07934e6i	1.00090	−	0.342190i
$209$	−149531.			−0.0163792
$210$		0			0
$211$	1.53834e6			0.163759			0.0818793	−	0.996642i	$-0.473908\pi$
$211$	1.53834e6			0.163759			0.0818793	+	0.996642i	$0.473908\pi$
$212$	−8.03713e6	−	5.74610e6i	−0.843516	−	0.603067i
$213$		0			0
$214$	−1.07746e7	+	5.54185e6i	−1.09941	+	0.565475i
$215$		−	1.08575e7i		−	1.09248i
$216$		0			0
$217$	−1.56813e7			−1.53463
$218$	4.98629e6	+	9.69449e6i	0.481291	+	0.935740i
$219$		0			0
$220$	839920.	+	600496.i	0.0788806	+	0.0563952i
$221$		−	1.75110e7i		−	1.62231i
$222$		0			0
$223$		−	9.74246e6i		−	0.878525i	−0.898359	−	0.439263i	$-0.855240\pi$
$223$		−	9.74246e6i		−	0.878525i	0.898359	−	0.439263i	$-0.144760\pi$
$224$	8.07854e6	−	8.39291e6i	0.718768	−	0.746738i
$225$		0			0
$226$	1.00453e6	−	516670.i	0.0870234	−	0.0447598i
$227$	−8.88934e6			−0.759962			−0.379981	−	0.924994i	$-0.624070\pi$
$227$	−8.88934e6			−0.759962			−0.379981	+	0.924994i	$0.624070\pi$
$228$		0			0
$229$			2.01802e6i			0.168042i	0.996464	+	0.0840211i	$0.0267763\pi$
$229$			2.01802e6i			0.168042i	−0.996464	+	0.0840211i	$0.973224\pi$
$230$	−9.39235e6	+	4.83088e6i	−0.771952	+	0.397048i
$231$		0			0
$232$	1.64611e7	+	2.41432e6i	1.31824	+	0.193344i
$233$	1.36525e7			1.07931			0.539654	−	0.841887i	$-0.318555\pi$
$233$	1.36525e7			1.07931			0.539654	+	0.841887i	$0.318555\pi$
$234$		0			0
$235$	−5.65634e6			−0.435845
$236$	−1.54633e6	+	2.16287e6i	−0.117643	+	0.164549i
$237$		0			0
$238$	−9.80183e6	−	1.90570e7i	−0.727070	−	1.41359i
$239$		−	2.35697e7i		−	1.72648i	−0.504797	−	0.863238i	$-0.668433\pi$
$239$		−	2.35697e7i		−	1.72648i	0.504797	−	0.863238i	$-0.331567\pi$
$240$		0			0
$241$	2.43293e7			1.73812			0.869059	−	0.494708i	$-0.164725\pi$
$241$	2.43293e7			1.73812			0.869059	+	0.494708i	$0.164725\pi$
$242$	1.23589e7	−	6.35670e6i	0.872034	−	0.448524i
$243$		0			0
$244$	5.64164e6	+	4.03346e6i	0.388361	+	0.277657i
$245$		−	760560.i		−	0.0517172i
$246$		0			0
$247$		−	1.87549e6i		−	0.124458i
$248$	2.23452e7	+	3.27733e6i	1.46497	+	0.214865i
$249$		0			0
$250$	−7.54085e6	−	1.46611e7i	−0.482614	−	0.938314i
$251$	−1.52915e7			−0.967006			−0.483503	−	0.875343i	$-0.660636\pi$
$251$	−1.52915e7			−0.967006			−0.483503	+	0.875343i	$0.660636\pi$
$252$		0			0
$253$		−	2.80946e6i		−	0.173485i
$254$	3.22017e6	+	6.26075e6i	0.196507	+	0.382054i
$255$		0			0
$256$	−1.32657e7	+	1.02712e7i	−0.790696	+	0.612209i
$257$	−1.86517e7			−1.09880			−0.549401	−	0.835559i	$-0.685144\pi$
$257$	−1.86517e7			−1.09880			−0.549401	+	0.835559i	$0.685144\pi$
$258$		0			0
$259$	−2.10080e7			−1.20917
$260$	−7.53170e6	+	1.05347e7i	−0.428522	+	0.599378i
$261$		0			0
$262$	−1.03157e7	+	5.30581e6i	−0.573582	+	0.295018i
$263$			1.48774e7i			0.817825i	0.912574	+	0.408913i	$0.134092\pi$
$263$			1.48774e7i			0.817825i	−0.912574	+	0.408913i	$0.865908\pi$
$264$		0			0
$265$	1.34414e7			0.722282
$266$	−1.04981e6	−	2.04107e6i	−0.0557784	−	0.108446i
$267$		0			0
$268$	1.79655e7	−	2.51285e7i	0.933328	−	1.30546i
$269$		−	5.66223e6i		−	0.290891i	−0.989366	−	0.145446i	$-0.953538\pi$
$269$		−	5.66223e6i		−	0.290891i	0.989366	−	0.145446i	$-0.0464616\pi$
$270$		0			0
$271$			4.23421e6i			0.212748i	0.994326	+	0.106374i	$0.0339240\pi$
$271$			4.23421e6i			0.212748i	−0.994326	+	0.106374i	$0.966076\pi$
$272$	9.98437e6	+	2.92040e7i	0.496151	+	1.45123i
$273$		0			0
$274$	3.04173e6	−	1.56449e6i	0.147866	−	0.0760540i
$275$	1.49039e6			0.0716641
$276$		0			0
$277$			1.44982e7i			0.682144i	0.940037	+	0.341072i	$0.110790\pi$
$277$			1.44982e7i			0.682144i	−0.940037	+	0.341072i	$0.889210\pi$
$278$	−1.15040e7	+	5.91700e6i	−0.535445	+	0.275402i
$279$		0			0
$280$	−2.29985e6	+	1.56807e7i	−0.104767	+	0.714316i
$281$	2.12914e7			0.959587			0.479794	−	0.877381i	$-0.340712\pi$
$281$	2.12914e7			0.959587			0.479794	+	0.877381i	$0.340712\pi$
$282$		0			0
$283$	−3.35684e6			−0.148105			−0.0740527	−	0.997254i	$-0.523593\pi$
$283$	−3.35684e6			−0.148105			−0.0740527	+	0.997254i	$0.523593\pi$
$284$	−2.17962e7	−	1.55831e7i	−0.951539	−	0.680297i
$285$		0			0
$286$	−1.57558e6	−	3.06328e6i	−0.0673506	−	0.130945i
$287$			3.09808e7i			1.31053i
$288$		0			0
$289$	3.26396e7			1.35223
$290$	−2.01282e7	+	1.03528e7i	−0.825296	+	0.424485i
$291$		0			0
$292$	−2.03289e7	+	2.84343e7i	−0.816518	+	1.14207i
$293$			3.28171e7i			1.30466i	0.757935	+	0.652330i	$0.226208\pi$
$293$			3.28171e7i			1.30466i	−0.757935	+	0.652330i	$0.773792\pi$
$294$		0			0
$295$		−	3.61721e6i		−	0.140899i
$296$	2.99356e7	+	4.39059e6i	1.15428	+	0.169296i
$297$		0			0
$298$	2.18181e7	+	4.24193e7i	0.824456	+	1.60293i
$299$	3.52375e7			1.31823
$300$		0			0
$301$			4.43308e7i			1.62557i
$302$	−2.48856e7	−	4.83834e7i	−0.903498	−	1.75661i
$303$		0			0
$304$	1.06936e6	+	3.12785e6i	0.0380630	+	0.111333i
$305$	−9.43515e6			−0.332544
$306$		0			0
$307$	−4.14283e7			−1.43180			−0.715899	−	0.698204i	$-0.753983\pi$
$307$	−4.14283e7			−1.43180			−0.715899	+	0.698204i	$0.753983\pi$
$308$	−3.42937e6	−	2.45180e6i	−0.117371	−	0.0839139i
$309$		0			0
$310$	−2.73230e7	+	1.40534e7i	−0.917158	+	0.471733i
$311$		−	2.67712e7i		−	0.889994i	−0.895532	−	0.444997i	$-0.853205\pi$
$311$		−	2.67712e7i		−	0.889994i	0.895532	−	0.444997i	$-0.146795\pi$
$312$		0			0
$313$	−1.82021e7			−0.593591			−0.296796	−	0.954941i	$-0.595918\pi$
$313$	−1.82021e7			−0.593591			−0.296796	+	0.954941i	$0.595918\pi$
$314$	−1.95694e7	−	3.80474e7i	−0.632104	−	1.22896i
$315$		0			0
$316$	2.80217e7	+	2.00340e7i	0.888043	+	0.634901i
$317$			3.55468e7i			1.11589i	0.829877	+	0.557946i	$0.188410\pi$
$317$			3.55468e7i			1.11589i	−0.829877	+	0.557946i	$0.811590\pi$
$318$		0			0
$319$		−	6.02077e6i		−	0.185473i
$320$	6.55439e6	−	2.18637e7i	0.200024	−	0.667226i
$321$		0			0
$322$	3.83486e7	−	1.97243e7i	1.14864	−	0.590792i
$323$	6.08102e6			0.180455
$324$		0			0
$325$			1.86932e7i			0.544544i
$326$	1.85220e6	−	952666.i	0.0534608	−	0.0274972i
$327$		0			0
$328$	6.47487e6	−	4.41464e7i	0.183489	−	1.25105i
$329$	2.30946e7			0.648520
$330$		0			0
$331$	1.32116e7			0.364311			0.182156	−	0.983270i	$-0.441693\pi$
$331$	1.32116e7			0.364311			0.182156	+	0.983270i	$0.441693\pi$
$332$	−1.65093e7	+	2.30918e7i	−0.451144	+	0.631020i
$333$		0			0
$334$	−1.00194e7	−	1.94800e7i	−0.268907	−	0.522818i
$335$			4.20252e7i			1.11783i
$336$		0			0
$337$	4.91286e7			1.28364			0.641822	−	0.766854i	$-0.278179\pi$
$337$	4.91286e7			1.28364			0.641822	+	0.766854i	$0.278179\pi$
$338$	4.08254e6	−	2.09983e6i	0.105726	−	0.0543792i
$339$		0			0
$340$	−3.41573e7	−	2.44206e7i	−0.869055	−	0.621326i
$341$		−	8.17292e6i		−	0.206117i
$342$		0			0
$343$		−	3.87195e7i		−	0.959505i
$344$	9.26496e6	−	6.31696e7i	0.227598	−	1.55179i
$345$		0			0
$346$	−2.14965e7	−	4.17941e7i	−0.518966	−	1.00899i
$347$	3.48521e7			0.834143			0.417072	−	0.908874i	$-0.363056\pi$
$347$	3.48521e7			0.834143			0.417072	+	0.908874i	$0.363056\pi$
$348$		0			0
$349$		−	4.31268e7i		−	1.01455i	−0.861786	−	0.507273i	$-0.830654\pi$
$349$		−	4.31268e7i		−	1.01455i	0.861786	−	0.507273i	$-0.169346\pi$
$350$	1.04636e7	+	2.03436e7i	0.244048	+	0.474486i
$351$		0			0
$352$	4.37429e6	+	4.21044e6i	0.100295	+	0.0965383i
$353$	−3.28423e7			−0.746637			−0.373318	−	0.927703i	$-0.621780\pi$
$353$	−3.28423e7			−0.746637			−0.373318	+	0.927703i	$0.621780\pi$
$354$		0			0
$355$	3.64523e7			0.814779
$356$	5.63682e6	−	7.88428e6i	0.124935	−	0.174748i
$357$		0			0
$358$	−8.59493e6	+	4.42074e6i	−0.187324	+	0.0963486i
$359$			2.12923e7i			0.460193i	0.973168	+	0.230097i	$0.0739042\pi$
$359$			2.12923e7i			0.460193i	−0.973168	+	0.230097i	$0.926096\pi$
$360$		0			0
$361$	−4.63946e7			−0.986156
$362$	1.35516e7	+	2.63474e7i	0.285670	+	0.555408i
$363$		0			0
$364$	3.07517e7	−	4.30127e7i	0.637624	−	0.891852i
$365$		−	4.75539e7i		−	0.977929i
$366$		0			0
$367$			4.13896e7i			0.837323i	0.908142	+	0.418662i	$0.137501\pi$
$367$			4.13896e7i			0.837323i	−0.908142	+	0.418662i	$0.862499\pi$
$368$	−5.87675e7	+	2.00916e7i	−1.17922	+	0.403155i
$369$		0			0
$370$	−3.66043e7	+	1.88271e7i	−0.722647	+	0.371688i
$371$	−5.48808e7			−1.07473
$372$		0			0
$373$		−	1.08054e7i		−	0.208217i	−0.994566	−	0.104108i	$-0.966801\pi$
$373$		−	1.08054e7i		−	0.208217i	0.994566	−	0.104108i	$-0.0331989\pi$
$374$	9.93230e6	−	5.10860e6i	0.189861	−	0.0976534i
$375$		0			0
$376$	−3.29089e7	−	4.82669e6i	−0.619085	−	0.0907999i
$377$	7.55153e7			1.40933
$378$		0			0
$379$	628401.			0.0115430			0.00577151	−	0.999983i	$-0.498163\pi$
$379$	628401.			0.0115430			0.00577151	+	0.999983i	$0.498163\pi$
$380$	−3.65837e6	−	2.61553e6i	−0.0666709	−	0.0476660i
$381$		0			0
$382$	5.22998e6	+	1.01683e7i	0.0938232	+	0.182414i
$383$		−	6.08976e7i		−	1.08394i	−0.840399	−	0.541969i	$-0.817679\pi$
$383$		−	6.08976e7i		−	1.08394i	0.840399	−	0.541969i	$-0.182321\pi$
$384$		0			0
$385$	5.73531e6			0.100502
$386$	8.76090e6	−	4.50610e6i	0.152330	−	0.0783500i
$387$		0			0
$388$	−4.40016e7	+	6.15455e7i	−0.753309	+	1.05366i
$389$		−	1.06270e7i		−	0.180535i	−0.995918	−	0.0902674i	$-0.971228\pi$
$389$		−	1.06270e7i		−	0.180535i	0.995918	−	0.0902674i	$-0.0287722\pi$
$390$		0			0
$391$			1.14253e8i			1.91134i
$392$	649004.	−	4.42498e6i	0.0107743	−	0.0734604i
$393$		0			0
$394$	−2.19907e7	−	4.27550e7i	−0.359543	−	0.699034i
$395$	−4.68639e7			−0.760409
$396$		0			0
$397$		−	8.47161e7i		−	1.35392i	−0.736018	−	0.676962i	$-0.763296\pi$
$397$		−	8.47161e7i		−	1.35392i	0.736018	−	0.676962i	$-0.236704\pi$
$398$	1.32702e7	+	2.58003e7i	0.210489	+	0.409238i
$399$		0			0
$400$	−1.06584e7	−	3.11756e7i	−0.166538	−	0.487119i
$401$	−2.73650e7			−0.424387			−0.212193	−	0.977228i	$-0.568061\pi$
$401$	−2.73650e7			−0.424387			−0.212193	+	0.977228i	$0.568061\pi$
$402$		0			0
$403$	1.02509e8			1.56619
$404$	−3.85965e6	−	2.75944e6i	−0.0585335	−	0.0418482i
$405$		0			0
$406$	8.21826e7	−	4.22700e7i	1.22801	−	0.631617i
$407$		−	1.09491e7i		−	0.162404i
$408$		0			0
$409$	−1.00483e8			−1.46867			−0.734336	−	0.678786i	$-0.762506\pi$
$409$	−1.00483e8			−1.46867			−0.734336	+	0.678786i	$0.762506\pi$
$410$	2.77646e7	+	5.39808e7i	0.402847	+	0.783228i
$411$		0			0
$412$	1.28548e7	+	9.19045e6i	0.183812	+	0.131415i
$413$			1.47690e7i			0.209652i
$414$		0			0
$415$		−	3.86190e7i		−	0.540327i
$416$	−5.28094e7	+	5.48644e7i	−0.733552	+	0.762097i
$417$		0			0
$418$	1.06378e6	−	547149.i	0.0145655	−	0.00749163i
$419$	−3.90978e7			−0.531508			−0.265754	−	0.964041i	$-0.585621\pi$
$419$	−3.90978e7			−0.531508			−0.265754	+	0.964041i	$0.585621\pi$
$420$		0			0
$421$			1.53254e7i			0.205383i	0.994713	+	0.102692i	$0.0327455\pi$
$421$			1.53254e7i			0.205383i	−0.994713	+	0.102692i	$0.967254\pi$
$422$	−1.09439e7	+	5.62893e6i	−0.145625	+	0.0749012i
$423$		0			0
$424$	7.82028e7	+	1.14699e7i	1.02595	+	0.150474i
$425$	−6.06102e7			−0.789548
$426$		0			0
$427$	3.85234e7			0.494813
$428$	5.63741e7	−	7.88510e7i	0.719032	−	1.00572i
$429$		0			0
$430$	3.97287e7	+	7.72418e7i	0.499688	+	0.971509i
$431$		−	2.62386e7i		−	0.327724i	−0.986483	−	0.163862i	$-0.947605\pi$
$431$		−	2.62386e7i		−	0.327724i	0.986483	−	0.163862i	$-0.0523952\pi$
$432$		0			0
$433$	−1.47614e8			−1.81830			−0.909149	−	0.416471i	$-0.863267\pi$
$433$	−1.47614e8			−1.81830			−0.909149	+	0.416471i	$0.863267\pi$
$434$	1.11559e8	−	5.73795e7i	1.36469	−	0.701920i
$435$		0			0
$436$	−7.09463e7	−	5.07226e7i	−0.855993	−	0.611987i
$437$			1.22369e7i			0.146631i
$438$		0			0
$439$		−	1.08726e8i		−	1.28511i	−0.766238	−	0.642557i	$-0.777874\pi$
$439$		−	1.08726e8i		−	1.28511i	0.766238	−	0.642557i	$-0.222126\pi$
$440$	−8.17259e6	−	1.19866e6i	−0.0959404	−	0.0140714i
$441$		0			0
$442$	6.40745e7	+	1.24576e8i	0.742025	+	1.44267i
$443$	1.36195e8			1.56657			0.783286	−	0.621662i	$-0.213542\pi$
$443$	1.36195e8			1.56657			0.783286	+	0.621662i	$0.213542\pi$
$444$		0			0
$445$			1.31858e7i			0.149632i
$446$	3.56487e7	+	6.93092e7i	0.401827	+	0.781244i
$447$		0			0
$448$	−2.67614e7	+	8.92685e7i	−0.297628	+	0.992806i
$449$	−3.60699e7			−0.398480			−0.199240	−	0.979951i	$-0.563847\pi$
$449$	−3.60699e7			−0.398480			−0.199240	+	0.979951i	$0.563847\pi$
$450$		0			0
$451$	−1.61468e7			−0.176018
$452$	−5.25579e6	+	7.35133e6i	−0.0569144	+	0.0796069i
$453$		0			0
$454$	6.32400e7	−	3.25270e7i	0.675810	−	0.347597i
$455$			7.19350e7i			0.763671i
$456$		0			0
$457$	−2.75944e7			−0.289117			−0.144558	−	0.989496i	$-0.546176\pi$
$457$	−2.75944e7			−0.289117			−0.144558	+	0.989496i	$0.546176\pi$
$458$	−7.38413e6	−	1.43564e7i	−0.0768604	−	0.149434i
$459$		0			0
$460$	4.91418e7	−	6.87351e7i	0.504867	−	0.706163i
$461$		−	2.35623e7i		−	0.240500i	−0.992744	−	0.120250i	$-0.961630\pi$
$461$		−	2.35623e7i		−	0.240500i	0.992744	−	0.120250i	$-0.0383697\pi$
$462$		0			0
$463$			9.92854e7i			1.00033i	0.865931	+	0.500164i	$0.166727\pi$
$463$			9.92854e7i			1.00033i	−0.865931	+	0.500164i	$0.833273\pi$
$464$	−1.25941e8	+	4.30571e7i	−1.26070	+	0.431014i
$465$		0			0
$466$	−9.71260e7	+	4.99560e7i	−0.959793	+	0.493662i
$467$	3.27967e7			0.322018			0.161009	−	0.986953i	$-0.448525\pi$
$467$	3.27967e7			0.322018			0.161009	+	0.986953i	$0.448525\pi$
$468$		0			0
$469$		−	1.71588e8i		−	1.66329i
$470$	4.02400e7	−	2.06971e7i	0.387583	−	0.199350i
$471$		0			0
$472$	3.08665e6	−	2.10452e7i	0.0293536	−	0.200136i
$473$	−2.31047e7			−0.218332
$474$		0			0
$475$	−6.49156e6			−0.0605714
$476$	1.39463e8	+	9.97084e7i	1.29312	+	0.924508i
$477$		0			0
$478$	8.62440e7	+	1.67678e8i	0.789669	+	1.53530i
$479$		−	4.77689e7i		−	0.434649i	−0.976099	−	0.217325i	$-0.930267\pi$
$479$		−	4.77689e7i		−	0.434649i	0.976099	−	0.217325i	$-0.0697330\pi$
$480$		0			0
$481$	1.37329e8			1.23404
$482$	−1.73082e8	+	8.90236e7i	−1.54565	+	0.794994i
$483$		0			0
$484$	−6.46631e7	+	9.04450e7i	−0.570322	+	0.797716i
$485$		−	1.02930e8i		−	0.902225i
$486$		0			0
$487$		−	1.08113e8i		−	0.936036i	−0.883719	−	0.468018i	$-0.844968\pi$
$487$		−	1.08113e8i		−	0.936036i	0.883719	−	0.468018i	$-0.155032\pi$
$488$	−5.48942e7	−	8.05123e6i	−0.472354	−	0.0692792i
$489$		0			0
$490$	2.78297e6	+	5.41073e6i	0.0236548	+	0.0459904i
$491$	3.35101e7			0.283095			0.141547	−	0.989931i	$-0.454792\pi$
$491$	3.35101e7			0.283095			0.141547	+	0.989931i	$0.454792\pi$
$492$		0			0
$493$			2.44849e8i			2.04342i
$494$	6.86260e6	+	1.33425e7i	0.0569256	+	0.110677i
$495$		0			0
$496$	−1.70959e8	+	5.84481e7i	−1.40103	+	0.478989i
$497$	−1.48834e8			−1.21236
$498$		0			0
$499$	−4.53985e6			−0.0365376			−0.0182688	−	0.999833i	$-0.505815\pi$
$499$	−4.53985e6			−0.0365376			−0.0182688	+	0.999833i	$0.505815\pi$
$500$	1.07293e8	+	7.67087e7i	0.858346	+	0.613670i
$501$		0			0
$502$	1.08786e8	−	5.59532e7i	0.859927	−	0.442297i
$503$		−	1.42040e8i		−	1.11611i	−0.829804	−	0.558055i	$-0.811548\pi$
$503$		−	1.42040e8i		−	1.11611i	0.829804	−	0.558055i	$-0.188452\pi$
$504$		0			0
$505$	6.45493e6			0.0501208
$506$	1.02801e7	+	1.99869e7i	0.0793497	+	0.154274i
$507$		0			0
$508$	−4.58174e7	−	3.27569e7i	−0.349494	−	0.249869i
$509$		−	8.91677e6i		−	0.0676167i	−0.999428	−	0.0338084i	$-0.989236\pi$
$509$		−	8.91677e6i		−	0.0676167i	0.999428	−	0.0338084i	$-0.0107636\pi$
$510$		0			0
$511$			1.94161e8i			1.45512i
$512$	5.67906e7	−	1.21611e8i	0.423123	−	0.906072i
$513$		0			0
$514$	1.32691e8	−	6.82485e7i	0.977128	−	0.502578i
$515$	−2.14985e7			−0.157393
$516$		0			0
$517$			1.20367e7i			0.0871032i
$518$	1.49454e8	−	7.68705e7i	1.07527	−	0.553058i
$519$		0			0
$520$	1.50341e7	−	1.02504e8i	0.106922	−	0.729008i
$521$	−8.65078e7			−0.611705			−0.305852	−	0.952079i	$-0.598941\pi$
$521$	−8.65078e7			−0.611705			−0.305852	+	0.952079i	$0.598941\pi$
$522$		0			0
$523$	6.65196e7			0.464991			0.232495	−	0.972598i	$-0.425311\pi$
$523$	6.65196e7			0.464991			0.232495	+	0.972598i	$0.425311\pi$
$524$	5.39729e7	−	7.54925e7i	0.375130	−	0.524699i
$525$		0			0
$526$	−5.44380e7	−	1.05840e8i	−0.374063	−	0.727265i
$527$			3.32371e8i			2.27086i
$528$		0			0
$529$	−8.18768e7			−0.553087
$530$	−9.56240e7	+	4.91835e7i	−0.642302	+	0.330363i
$531$		0			0
$532$	1.49370e7	+	1.06791e7i	0.0992038	+	0.0709251i
$533$		−	2.02521e8i		−	1.33749i
$534$		0			0
$535$			1.31871e8i			0.861171i
$536$	−3.58611e7	+	2.44505e8i	−0.232878	+	1.58779i
$537$		0			0
$538$	2.07187e7	+	4.02819e7i	0.133050	+	0.258680i
$539$	−1.61847e6			−0.0103356
$540$		0			0
$541$		−	1.61762e8i		−	1.02161i	−0.859696	−	0.510806i	$-0.829347\pi$
$541$		−	1.61762e8i		−	1.02161i	0.859696	−	0.510806i	$-0.170653\pi$
$542$	−1.54934e7	−	3.01228e7i	−0.0973082	−	0.189190i
$543$		0			0
$544$	−1.77891e8	−	1.71228e8i	−1.10499	−	1.06360i
$545$	1.18651e8			0.732965
$546$		0			0
$547$	−2.25695e8			−1.37899			−0.689495	−	0.724291i	$-0.742167\pi$
$547$	−2.25695e8			−1.37899			−0.689495	+	0.724291i	$0.742167\pi$
$548$	−1.59147e7	+	2.22600e7i	−0.0967066	+	0.135265i
$549$		0			0
$550$	−1.06028e7	+	5.45349e6i	−0.0637285	+	0.0327783i
$551$			2.62241e7i			0.156764i
$552$		0			0
$553$	1.91344e8			1.13146
$554$	−5.30506e7	−	1.03143e8i	−0.312004	−	0.606608i
$555$		0			0
$556$	6.01902e7	−	8.41887e7i	0.350188	−	0.489812i
$557$			2.35751e8i			1.36423i	0.731244	+	0.682116i	$0.238940\pi$
$557$			2.35751e8i			1.36423i	−0.731244	+	0.682116i	$0.761060\pi$
$558$		0			0
$559$		−	2.89790e8i		−	1.65901i
$560$	−4.10157e7	−	1.19970e8i	−0.233553	−	0.683138i
$561$		0			0
$562$	−1.51470e8	+	7.79073e7i	−0.853330	+	0.438904i
$563$	−1.47965e8			−0.829151			−0.414575	−	0.910015i	$-0.636070\pi$
$563$	−1.47965e8			−0.829151			−0.414575	+	0.910015i	$0.636070\pi$
$564$		0			0
$565$		−	1.22945e7i		−	0.0681654i
$566$	2.38810e7	−	1.22830e7i	0.131705	−	0.0677416i
$567$		0			0
$568$	2.12082e8	+	3.11056e7i	1.15733	+	0.169744i
$569$	5.13986e7			0.279007			0.139503	−	0.990222i	$-0.455449\pi$
$569$	5.13986e7			0.279007			0.139503	+	0.990222i	$0.455449\pi$
$570$		0			0
$571$	1.93179e8			1.03765			0.518826	−	0.854880i	$-0.326369\pi$
$571$	1.93179e8			1.03765			0.518826	+	0.854880i	$0.326369\pi$
$572$	2.24177e7	+	1.60274e7i	0.119785	+	0.0856398i
$573$		0			0
$574$	−1.13362e8	−	2.20402e8i	−0.599421	−	1.16541i
$575$		−	1.21966e8i		−	0.641559i
$576$		0			0
$577$	−6.54832e7			−0.340881			−0.170440	−	0.985368i	$-0.554519\pi$
$577$	−6.54832e7			−0.340881			−0.170440	+	0.985368i	$0.554519\pi$
$578$	−2.32202e8	+	1.19431e8i	−1.20249	+	0.618494i
$579$		0			0
$580$	1.05313e8	−	1.47302e8i	0.539755	−	0.754961i
$581$			1.57680e8i			0.803986i
$582$		0			0
$583$		−	2.86032e7i		−	0.144348i
$584$	4.05788e7	−	2.76671e8i	0.203733	−	1.38907i
$585$		0			0
$586$	−1.20081e8	−	2.33465e8i	−0.596735	−	1.16019i
$587$	2.92379e8			1.44554			0.722772	−	0.691087i	$-0.242868\pi$
$587$	2.92379e8			1.44554			0.722772	+	0.691087i	$0.242868\pi$
$588$		0			0
$589$			3.55981e7i			0.174213i
$590$	1.32358e7	+	2.57334e7i	0.0644455	+	0.125297i
$591$		0			0
$592$	−2.29031e8	+	7.83020e7i	−1.10390	+	0.377405i
$593$	3.63949e8			1.74533			0.872663	−	0.488323i	$-0.162391\pi$
$593$	3.63949e8			1.74533			0.872663	+	0.488323i	$0.162391\pi$
$594$		0			0
$595$	−2.33240e8			−1.10727
$596$	−3.10433e8	−	2.21942e8i	−1.46632	−	1.04834i
$597$		0			0
$598$	−2.50685e8	+	1.28938e8i	−1.17226	+	0.602943i
$599$		−	1.03797e8i		−	0.482952i	−0.970407	−	0.241476i	$-0.922368\pi$
$599$		−	1.03797e8i		−	0.482952i	0.970407	−	0.241476i	$-0.0776315\pi$
$600$		0			0
$601$	1.12596e8			0.518678			0.259339	−	0.965786i	$-0.416495\pi$
$601$	1.12596e8			0.518678			0.259339	+	0.965786i	$0.416495\pi$
$602$	−1.62211e8	−	3.15375e8i	−0.743517	−	1.44557i
$603$		0			0
$604$	3.54079e8	+	2.53147e8i	1.60690	+	1.14885i
$605$		−	1.51261e8i		−	0.683064i
$606$		0			0
$607$		−	1.30862e8i		−	0.585124i	−0.956247	−	0.292562i	$-0.905492\pi$
$607$		−	1.30862e8i		−	0.585124i	0.956247	−	0.292562i	$-0.0945078\pi$
$608$	−1.90527e7	−	1.83391e7i	−0.0847707	−	0.0815955i
$609$		0			0
$610$	6.71230e7	−	3.45242e7i	0.295721	−	0.152102i
$611$	−1.50969e8			−0.661859
$612$		0			0
$613$			9.03522e7i			0.392245i	0.980579	+	0.196122i	$0.0628350\pi$
$613$			9.03522e7i			0.392245i	−0.980579	+	0.196122i	$0.937165\pi$
$614$	2.94726e8	−	1.51590e8i	1.27325	−	0.654887i
$615$		0			0
$616$	3.33684e7	+	4.89408e6i	0.142756	+	0.0209377i
$617$	5.06545e7			0.215656			0.107828	−	0.994170i	$-0.465610\pi$
$617$	5.06545e7			0.215656			0.107828	+	0.994170i	$0.465610\pi$
$618$		0			0
$619$	3.21361e8			1.35494			0.677472	−	0.735549i	$-0.263076\pi$
$619$	3.21361e8			1.35494			0.677472	+	0.735549i	$0.263076\pi$
$620$	1.42957e8	−	1.99956e8i	0.599833	−	0.838993i
$621$		0			0
$622$	9.79586e7	+	1.90454e8i	0.407072	+	0.791442i
$623$		−	5.38371e7i		−	0.222647i
$624$		0			0
$625$	−5.37550e7			−0.220180
$626$	1.29492e8	−	6.66032e7i	0.527861	−	0.271501i
$627$		0			0
$628$	2.78439e8	+	1.99068e8i	1.12422	+	0.803753i
$629$			4.45272e8i			1.78926i
$630$		0			0
$631$		−	4.01150e6i		−	0.0159668i	−0.999968	−	0.00798341i	$-0.997459\pi$
$631$		−	4.01150e6i		−	0.0159668i	0.999968	−	0.00798341i	$-0.00254123\pi$
$632$	−2.72657e8	−	3.99901e7i	−1.08010	−	0.158417i
$633$		0			0
$634$	−1.30069e8	−	2.52885e8i	−0.510396	−	0.992327i
$635$	7.66256e7			0.299263
$636$		0			0
$637$		−	2.02996e7i		−	0.0785360i
$638$	2.20306e7	+	4.28326e7i	0.0848330	+	0.164935i
$639$		0			0
$640$	3.33725e7	+	1.79524e8i	0.127306	+	0.684831i
$641$	1.44455e8			0.548476			0.274238	−	0.961662i	$-0.411574\pi$
$641$	1.44455e8			0.548476			0.274238	+	0.961662i	$0.411574\pi$
$642$		0			0
$643$	−5.52753e7			−0.207921			−0.103961	−	0.994581i	$-0.533152\pi$
$643$	−5.52753e7			−0.207921			−0.103961	+	0.994581i	$0.533152\pi$
$644$	−2.00644e8	+	2.80643e8i	−0.751223	+	1.05074i
$645$		0			0
$646$	−4.32612e7	+	2.22511e7i	−0.160473	+	0.0825380i
$647$			1.66453e8i			0.614579i	0.951616	+	0.307290i	$0.0994220\pi$
$647$			1.66453e8i			0.614579i	−0.951616	+	0.307290i	$0.900578\pi$
$648$		0			0
$649$	−7.69741e6			−0.0281586
$650$	−6.84002e7	−	1.32986e8i	−0.249068	−	0.484245i
$651$		0			0
$652$	−9.69092e6	+	1.35548e7i	−0.0349641	+	0.0489047i
$653$			4.66347e8i			1.67483i	0.546571	+	0.837413i	$0.315933\pi$
$653$			4.66347e8i			1.67483i	−0.546571	+	0.837413i	$0.684067\pi$
$654$		0			0
$655$			1.26255e8i			0.449287i
$656$	1.15473e8	+	3.37756e8i	0.409043	+	1.19644i
$657$		0			0
$658$	−1.64298e8	+	8.45057e7i	−0.576708	+	0.296625i
$659$	−3.34752e8			−1.16968			−0.584840	−	0.811149i	$-0.698843\pi$
$659$	−3.34752e8			−1.16968			−0.584840	+	0.811149i	$0.698843\pi$
$660$		0			0
$661$			4.98794e8i			1.72710i	0.504266	+	0.863548i	$0.331763\pi$
$661$			4.98794e8i			1.72710i	−0.504266	+	0.863548i	$0.668237\pi$
$662$	−9.39894e7	+	4.83427e7i	−0.323970	+	0.166632i
$663$		0			0
$664$	3.29545e7	−	2.24688e8i	0.112567	−	0.767494i
$665$	−2.49808e7			−0.0849457
$666$		0			0
$667$	−4.92712e8			−1.66041
$668$	1.42559e8	+	1.01922e8i	0.478261	+	0.341930i
$669$		0			0
$670$	−1.53775e8	−	2.98973e8i	−0.511282	−	0.994049i
$671$			2.00779e7i			0.0664587i
$672$		0			0
$673$	2.44999e6			0.00803748			0.00401874	−	0.999992i	$-0.498721\pi$
$673$	2.44999e6			0.00803748			0.00401874	+	0.999992i	$0.498721\pi$
$674$	−3.49508e8	+	1.79767e8i	−1.14150	+	0.587123i
$675$		0			0
$676$	−2.13603e7	+	2.98769e7i	−0.0691461	+	0.0967153i
$677$		−	2.33674e8i		−	0.753084i	−0.926400	−	0.376542i	$-0.877113\pi$
$677$		−	2.33674e8i		−	0.753084i	0.926400	−	0.376542i	$-0.122887\pi$
$678$		0			0
$679$			4.20258e8i			1.34248i
$680$	3.32357e8	+	4.87462e7i	1.05701	+	0.155029i
$681$		0			0
$682$	2.99056e7	+	5.81433e7i	0.0942755	+	0.183293i
$683$	−4.82431e8			−1.51417			−0.757083	−	0.653319i	$-0.773376\pi$
$683$	−4.82431e8			−1.51417			−0.757083	+	0.653319i	$0.773376\pi$
$684$		0			0
$685$		−	3.72280e7i		−	0.115824i
$686$	1.41679e8	+	2.75456e8i	0.438866	+	0.853257i
$687$		0			0
$688$	1.65232e8	+	4.83298e8i	0.507374	+	1.48406i
$689$	3.58755e8			1.09683
$690$		0			0
$691$	7.81246e6			0.0236785			0.0118392	−	0.999930i	$-0.496231\pi$
$691$	7.81246e6			0.0236785			0.0118392	+	0.999930i	$0.496231\pi$
$692$	3.05858e8	+	2.18671e8i	0.922999	+	0.659893i
$693$		0			0
$694$	−2.47943e8	+	1.27527e8i	−0.741776	+	0.381527i
$695$			1.40798e8i			0.419414i
$696$		0			0
$697$	6.56650e8			1.93926
$698$	1.57806e8	+	3.06810e8i	0.464041	+	0.902202i
$699$		0			0
$700$	−1.48878e8	−	1.06440e8i	−0.434048	−	0.310320i
$701$		−	4.10282e8i		−	1.19104i	−0.803339	−	0.595522i	$-0.796945\pi$
$701$		−	4.10282e8i		−	1.19104i	0.803339	−	0.595522i	$-0.203055\pi$
$702$		0			0
$703$			4.76902e7i			0.137266i
$704$	−4.65257e7	−	1.39477e7i	−0.133345	−	0.0399747i
$705$		0			0
$706$	2.33645e8	−	1.20173e8i	0.663960	−	0.341503i
$707$	−2.63553e7			−0.0745778
$708$		0			0
$709$		−	2.31730e8i		−	0.650194i	−0.945681	−	0.325097i	$-0.894603\pi$
$709$		−	2.31730e8i		−	0.650194i	0.945681	−	0.325097i	$-0.105397\pi$
$710$	−2.59327e8	+	1.33383e8i	−0.724557	+	0.372670i
$711$		0			0
$712$	−1.12517e7	+	7.67156e7i	−0.0311730	+	0.212541i
$713$	−6.68833e8			−1.84522
$714$		0			0
$715$	−3.74917e7			−0.102569
$716$	4.49696e7	−	6.28995e7i	0.122512	−	0.171359i
$717$		0			0
$718$	−7.79109e7	−	1.51477e8i	−0.210487	−	0.409235i
$719$		−	5.75272e8i		−	1.54770i	−0.633370	−	0.773849i	$-0.718329\pi$
$719$		−	5.75272e8i		−	1.54770i	0.633370	−	0.773849i	$-0.281671\pi$
$720$		0			0
$721$	8.77777e7			0.234195
$722$	3.30057e8	−	1.69763e8i	0.876956	−	0.451056i
$723$		0			0
$724$	−1.92816e8	−	1.37853e8i	−0.508074	−	0.363244i
$725$		−	2.61379e8i		−	0.685892i
$726$		0			0
$727$		−	4.29564e8i		−	1.11796i	−0.829182	−	0.558978i	$-0.811194\pi$
$727$		−	4.29564e8i		−	1.11796i	0.829182	−	0.558978i	$-0.188806\pi$
$728$	−6.13838e7	+	4.18522e8i	−0.159096	+	1.08474i
$729$		0			0
$730$	1.74004e8	+	3.38305e8i	0.447293	+	0.869640i
$731$	9.39607e8			2.40544
$732$		0			0
$733$			1.88436e8i			0.478467i	0.970962	+	0.239233i	$0.0768961\pi$
$733$			1.88436e8i			0.478467i	−0.970962	+	0.239233i	$0.923104\pi$
$734$	−1.51449e8	−	2.94451e8i	−0.382981	−	0.744604i
$735$		0			0
$736$	3.44563e8	−	3.57971e8i	0.864241	−	0.897872i
$737$	8.94294e7			0.223397
$738$		0			0
$739$	−2.19850e8			−0.544745			−0.272372	−	0.962192i	$-0.587808\pi$
$739$	−2.19850e8			−0.544745			−0.272372	+	0.962192i	$0.587808\pi$
$740$	1.91517e8	−	2.67877e8i	0.472621	−	0.661060i
$741$		0			0
$742$	3.90429e8	−	2.00814e8i	0.955720	−	0.491567i
$743$			2.54455e7i			0.0620360i	0.999519	+	0.0310180i	$0.00987493\pi$
$743$			2.54455e7i			0.0620360i	−0.999519	+	0.0310180i	$0.990125\pi$
$744$		0			0
$745$	5.19173e8			1.25558
$746$	3.95382e7	+	7.68714e7i	0.0952359	+	0.185160i
$747$		0			0
$748$	−5.19669e7	+	7.26866e7i	−0.124171	+	0.173680i
$749$		−	5.38427e8i		−	1.28139i
$750$		0			0
$751$		−	5.69177e8i		−	1.34378i	−0.740652	−	0.671889i	$-0.765483\pi$
$751$		−	5.69177e8i		−	1.34378i	0.740652	−	0.671889i	$-0.234517\pi$
$752$	2.51780e8	−	8.60794e7i	0.592062	−	0.202416i
$753$		0			0
$754$	−5.37227e8	+	2.76318e8i	−1.25327	+	0.644608i
$755$	−5.92167e8			−1.37595
$756$		0			0
$757$			8.99806e7i			0.207425i	0.994607	+	0.103713i	$0.0330722\pi$
$757$			8.99806e7i			0.207425i	−0.994607	+	0.103713i	$0.966928\pi$
$758$	−4.47053e6	+	2.29939e6i	−0.0102648	+	0.00527964i
$759$		0			0
$760$	3.55966e7	+	5.22089e6i	0.0810901	+	0.0118933i
$761$	−4.53947e8			−1.03003			−0.515016	−	0.857180i	$-0.672214\pi$
$761$	−4.53947e8			−1.03003			−0.515016	+	0.857180i	$0.672214\pi$
$762$		0			0
$763$	−4.84450e8			−1.09062
$764$	−7.44137e7	−	5.32016e7i	−0.166868	−	0.119301i
$765$		0			0
$766$	2.22831e8	+	4.33234e8i	0.495780	+	0.963910i
$767$		−	9.65446e7i		−	0.213965i
$768$		0			0
$769$	−7.66522e8			−1.68557			−0.842783	−	0.538254i	$-0.819084\pi$
$769$	−7.66522e8			−1.68557			−0.842783	+	0.538254i	$0.819084\pi$
$770$	−4.08018e7	+	2.09861e7i	−0.0893732	+	0.0459684i
$771$		0			0
$772$	−4.58379e7	+	6.41140e7i	−0.0996261	+	0.139348i
$773$			8.01785e7i			0.173588i	0.996226	+	0.0867939i	$0.0276622\pi$
$773$			8.01785e7i			0.173588i	−0.996226	+	0.0867939i	$0.972338\pi$
$774$		0			0
$775$		−	3.54809e8i		−	0.762237i
$776$	8.78321e7	−	5.98850e8i	0.187961	−	1.28154i
$777$		0			0
$778$	3.88852e7	+	7.56018e7i	0.0825744	+	0.160544i
$779$	7.03294e7			0.148773
$780$		0			0
$781$		−	7.75703e7i		−	0.162833i
$782$	−4.18064e8	−	8.12813e8i	−0.874223	−	1.69969i
$783$		0			0
$784$	1.15744e7	+	3.38547e7i	0.0240187	+	0.0702540i
$785$	−4.65664e8			−0.962640
$786$		0			0
$787$	−2.53600e8			−0.520265			−0.260132	−	0.965573i	$-0.583766\pi$
$787$	−2.53600e8			−0.520265			−0.260132	+	0.965573i	$0.583766\pi$
$788$	3.12890e8	+	2.23699e8i	0.639459	+	0.457178i
$789$		0			0
$790$	3.33396e8	−	1.71480e8i	0.676207	−	0.347802i
$791$			5.01978e7i			0.101427i
$792$		0			0
$793$	−2.51827e8			−0.504990
$794$	3.09985e8	+	6.02682e8i	0.619269	+	1.20400i
$795$		0			0
$796$	−1.88812e8	−	1.34990e8i	−0.374361	−	0.267647i
$797$		−	2.34122e8i		−	0.462453i	−0.972900	−	0.231226i	$-0.925726\pi$
$797$		−	2.34122e8i		−	0.462453i	0.972900	−	0.231226i	$-0.0742738\pi$
$798$		0			0
$799$		−	4.89499e8i		−	0.959647i
$800$	1.89900e8	+	1.82787e8i	0.370899	+	0.357006i
$801$		0			0
$802$	1.94678e8	−	1.00131e8i	0.377393	−	0.194109i
$803$	−1.01194e8			−0.195438
$804$		0			0
$805$		−	4.69351e8i		−	0.899726i
$806$	−7.29260e8	+	3.75090e8i	−1.39276	+	0.716357i
$807$		0			0
$808$	3.75552e7	+	5.50815e6i	0.0711928	+	0.0104417i
$809$	7.71750e8			1.45757			0.728787	−	0.684740i	$-0.240084\pi$
$809$	7.71750e8			1.45757			0.728787	+	0.684740i	$0.240084\pi$
$810$		0			0
$811$	7.06092e8			1.32373			0.661864	−	0.749624i	$-0.269766\pi$
$811$	7.06092e8			1.32373			0.661864	+	0.749624i	$0.269766\pi$
$812$	−4.29988e8	+	6.01429e8i	−0.803134	+	1.12335i
$813$		0			0
$814$	4.00640e7	+	7.78937e7i	0.0742816	+	0.144421i
$815$		−	2.26692e7i		−	0.0418759i
$816$		0			0
$817$	1.00635e8			0.184537
$818$	7.14853e8	−	3.67679e8i	1.30604	−	0.671753i
$819$		0			0
$820$	−3.95043e8	−	2.82434e8i	−0.716478	−	0.512241i
$821$			7.89885e8i			1.42736i	0.700470	+	0.713682i	$0.252974\pi$
$821$			7.89885e8i			1.42736i	−0.700470	+	0.713682i	$0.747026\pi$
$822$		0			0
$823$			3.99325e8i			0.716353i	0.933654	+	0.358177i	$0.116601\pi$
$823$			3.99325e8i			0.716353i	−0.933654	+	0.358177i	$0.883399\pi$
$824$	−1.25080e8	−	1.83452e7i	−0.223565	−	0.0327899i
$825$		0			0
$826$	−5.40412e7	−	1.05068e8i	−0.0958925	−	0.186437i
$827$	9.56996e8			1.69197			0.845987	−	0.533204i	$-0.179012\pi$
$827$	9.56996e8			1.69197			0.845987	+	0.533204i	$0.179012\pi$
$828$		0			0
$829$		−	8.07615e8i		−	1.41756i	−0.705431	−	0.708779i	$-0.749246\pi$
$829$		−	8.07615e8i		−	1.41756i	0.705431	−	0.708779i	$-0.250754\pi$
$830$	1.41311e8	+	2.74741e8i	0.247139	+	0.480495i
$831$		0			0
$832$	1.74939e8	−	5.83548e8i	0.303750	−	1.01323i
$833$	6.58188e7			0.113871
$834$		0			0
$835$	−2.38417e8			−0.409523
$836$	−5.56583e6	+	7.78498e6i	−0.00952601	+	0.0133241i
$837$		0			0
$838$	2.78147e8	−	1.43063e8i	0.472653	−	0.243105i
$839$			2.45137e8i			0.415072i	0.978227	+	0.207536i	$0.0665445\pi$
$839$			2.45137e8i			0.415072i	−0.978227	+	0.207536i	$0.933456\pi$
$840$		0			0
$841$	−4.61076e8			−0.775148
$842$	−5.60772e7	−	1.09027e8i	−0.0939399	−	0.182641i
$843$		0			0
$844$	5.72599e7	−	8.00900e7i	0.0952408	−	0.133214i
$845$		−	4.99665e7i		−	0.0828149i
$846$		0			0
$847$			6.17595e8i			1.01637i
$848$	−5.98315e8	+	2.04554e8i	−0.981166	+	0.335444i
$849$		0			0
$850$	4.31189e8	−	2.21779e8i	0.702119	−	0.361130i
$851$	−8.96026e8			−1.45389
$852$		0			0
$853$		−	4.49652e8i		−	0.724484i	−0.932084	−	0.362242i	$-0.882011\pi$
$853$		−	4.49652e8i		−	0.724484i	0.932084	−	0.362242i	$-0.117989\pi$
$854$	−2.74061e8	+	1.40961e8i	−0.440021	+	0.226321i
$855$		0			0
$856$	−1.12529e8	+	7.67236e8i	−0.179409	+	1.22323i
$857$	−1.20139e9			−1.90872			−0.954358	−	0.298664i	$-0.903459\pi$
$857$	−1.20139e9			−1.90872			−0.954358	+	0.298664i	$0.903459\pi$
$858$		0			0
$859$	8.99649e8			1.41936			0.709682	−	0.704523i	$-0.248839\pi$
$859$	8.99649e8			1.41936			0.709682	+	0.704523i	$0.248839\pi$
$860$	−5.65271e8	−	4.04137e8i	−0.888713	−	0.635380i
$861$		0			0
$862$	9.60097e7	+	1.86665e8i	0.149897	+	0.291435i
$863$			2.63559e8i			0.410059i	0.978756	+	0.205029i	$0.0657290\pi$
$863$			2.63559e8i			0.410059i	−0.978756	+	0.205029i	$0.934271\pi$
$864$		0			0
$865$	−5.11520e8			−0.790341
$866$	1.05015e9	−	5.40136e8i	1.61695	−	0.831667i
$867$		0			0
$868$	−5.83689e8	+	8.16412e8i	−0.892529	+	1.24839i
$869$			9.97261e7i			0.151967i
$870$		0			0
$871$			1.12167e9i			1.69750i
$872$	6.90321e8	+	1.01248e8i	1.04112	+	0.152699i
$873$		0			0
$874$	−4.47761e7	−	8.70550e7i	−0.0670674	−	0.130395i
$875$	7.32642e8			1.09362
$876$		0			0
$877$			1.13280e9i			1.67940i	0.543048	+	0.839702i	$0.317270\pi$
$877$			1.13280e9i			1.67940i	−0.543048	+	0.839702i	$0.682730\pi$
$878$	3.97841e8	+	7.73494e8i	0.587795	+	1.14281i
$879$		0			0
$880$	6.25269e7	−	2.13769e7i	0.0917527	−	0.0313687i
$881$	9.92597e8			1.45159			0.725797	−	0.687908i	$-0.241471\pi$
$881$	9.92597e8			1.45159			0.725797	+	0.687908i	$0.241471\pi$
$882$		0			0
$883$	6.02184e8			0.874676			0.437338	−	0.899297i	$-0.355921\pi$
$883$	6.02184e8			0.874676			0.437338	+	0.899297i	$0.355921\pi$
$884$	−9.11670e8	−	6.51793e8i	−1.31972	−	0.943524i
$885$		0			0
$886$	−9.68911e8	+	4.98352e8i	−1.39310	+	0.716531i
$887$		−	9.81438e8i		−	1.40635i	−0.711019	−	0.703173i	$-0.751766\pi$
$887$		−	9.81438e8i		−	1.40635i	0.711019	−	0.703173i	$-0.248234\pi$
$888$		0			0
$889$	−3.12860e8			−0.445292
$890$	−4.82481e7	−	9.38054e7i	−0.0684400	−	0.133063i
$891$		0			0
$892$	−5.07219e8	−	3.62633e8i	−0.714663	−	0.510944i
$893$		−	5.24270e7i		−	0.0736209i
$894$		0			0
$895$			1.05194e8i			0.146731i
$896$	−1.36259e8	−	7.32991e8i	−0.189426	−	1.01900i
$897$		0			0
$898$	2.56606e8	−	1.31984e8i	0.354355	−	0.182260i
$899$	−1.43333e9			−1.97273
$900$		0			0
$901$			1.16322e9i			1.59033i
$902$	1.14871e8	−	5.90830e7i	0.156527	−	0.0805087i
$903$		0			0
$904$	1.04911e7	−	7.15298e7i	0.0142010	−	0.0968238i
$905$	3.22468e8			0.435051
$906$		0			0
$907$	1.42352e9			1.90784			0.953919	−	0.300064i	$-0.0970081\pi$
$907$	1.42352e9			1.90784			0.953919	+	0.300064i	$0.0970081\pi$
$908$	−3.30878e8	+	4.62803e8i	−0.441989	+	0.618214i
$909$		0			0
$910$	−2.63218e8	−	5.11756e8i	−0.349294	−	0.679107i
$911$			1.18433e9i			1.56645i	0.621735	+	0.783227i	$0.286428\pi$
$911$			1.18433e9i			1.56645i	−0.621735	+	0.783227i	$0.713572\pi$
$912$		0			0
$913$	−8.21810e7			−0.107984
$914$	1.96311e8	−	1.00971e8i	0.257102	−	0.132238i
$915$		0			0
$916$	1.05063e8	+	7.51145e7i	0.136699	+	0.0977321i
$917$		−	5.15494e8i		−	0.668521i
$918$		0			0
$919$			4.58083e8i			0.590197i	0.955467	+	0.295099i	$0.0953525\pi$
$919$			4.58083e8i			0.590197i	−0.955467	+	0.295099i	$0.904647\pi$
$920$	−9.80924e7	+	6.68806e8i	−0.125971	+	0.858888i
$921$		0			0
$922$	8.62170e7	+	1.67626e8i	0.110002	+	0.213869i
$923$	9.72923e8			1.23730
$924$		0			0
$925$		−	4.75333e8i		−	0.600582i
$926$	−3.63296e8	−	7.06330e8i	−0.457538	−	0.889559i
$927$		0			0
$928$	7.38411e8	−	7.67146e8i	0.923963	−	0.959917i
$929$	−1.80879e8			−0.225600			−0.112800	−	0.993618i	$-0.535982\pi$
$929$	−1.80879e8			−0.225600			−0.112800	+	0.993618i	$0.535982\pi$
$930$		0			0
$931$	7.04942e6			0.00873583
$932$	5.08174e8	−	7.10788e8i	0.627718	−	0.877996i
$933$		0			0
$934$	−2.33321e8	+	1.20007e8i	−0.286360	+	0.147287i
$935$		−	1.21562e8i		−	0.148718i
$936$		0			0
$937$	4.41475e8			0.536645			0.268323	−	0.963329i	$-0.413531\pi$
$937$	4.41475e8			0.536645			0.268323	+	0.963329i	$0.413531\pi$
$938$	6.27856e8	+	1.22070e9i	0.760767	+	1.47911i
$939$		0			0
$940$	−2.10540e8	+	2.94484e8i	−0.253484	+	0.354551i
$941$			7.93413e8i			0.952205i	0.879390	+	0.476103i	$0.157951\pi$
$941$			7.93413e8i			0.952205i	−0.879390	+	0.476103i	$0.842049\pi$
$942$		0			0
$943$			1.32138e9i			1.57577i
$944$	5.50475e7	+	1.61013e8i	0.0654368	+	0.191401i
$945$		0			0
$946$	1.64370e8	−	8.45425e7i	0.194155	−	0.0998623i
$947$	1.05871e9			1.24660			0.623301	−	0.781982i	$-0.285791\pi$
$947$	1.05871e9			1.24660			0.623301	+	0.781982i	$0.285791\pi$
$948$		0			0
$949$		−	1.26923e9i		−	1.48505i
$950$	4.61818e7	−	2.37533e7i	0.0538642	−	0.0277046i
$951$		0			0
$952$	−1.35700e9	−	1.99029e8i	−1.57279	−	0.230678i
$953$	7.19443e8			0.831223			0.415611	−	0.909542i	$-0.363568\pi$
$953$	7.19443e8			0.831223			0.415611	+	0.909542i	$0.363568\pi$
$954$		0			0
$955$	1.24450e8			0.142885
$956$	−1.22710e9	−	8.77310e8i	−1.40445	−	1.00411i
$957$		0			0
$958$	1.74791e8	+	3.39835e8i	0.198803	+	0.386519i
$959$			1.52000e8i			0.172341i
$960$		0			0
$961$	−1.05818e9			−1.19231
$962$	−9.76979e8	+	5.02502e8i	−1.09739	+	0.564433i
$963$		0			0
$964$	9.05585e8	−	1.26665e9i	1.01088	−	1.41392i
$965$		−	1.07225e8i		−	0.119320i
$966$		0			0
$967$		−	1.39553e9i		−	1.54333i	−0.636029	−	0.771665i	$-0.719424\pi$
$967$		−	1.39553e9i		−	1.54333i	0.636029	−	0.771665i	$-0.280576\pi$
$968$	1.29075e8	−	8.80047e8i	0.142303	−	0.970241i
$969$		0			0
$970$	3.76630e8	+	7.32255e8i	0.412667	+	0.802319i
$971$	−2.27166e8			−0.248133			−0.124067	−	0.992274i	$-0.539594\pi$
$971$	−2.27166e8			−0.248133			−0.124067	+	0.992274i	$0.539594\pi$
$972$		0			0
$973$		−	5.74874e8i		−	0.624072i
$974$	3.95598e8	+	7.69133e8i	0.428131	+	0.832386i
$975$		0			0
$976$	4.19985e8	−	1.43586e8i	0.451736	−	0.154441i
$977$	−4.53866e8			−0.486680			−0.243340	−	0.969941i	$-0.578243\pi$
$977$	−4.53866e8			−0.486680			−0.243340	+	0.969941i	$0.578243\pi$
$978$		0			0
$979$	2.80593e7			0.0299039
$980$	−3.95968e7	−	2.83095e7i	−0.0420709	−	0.0300784i
$981$		0			0
$982$	−2.38396e8	+	1.22617e8i	−0.251747	+	0.129484i
$983$			1.59160e9i			1.67561i	0.545969	+	0.837806i	$0.316162\pi$
$983$			1.59160e9i			1.67561i	−0.545969	+	0.837806i	$0.683838\pi$
$984$		0			0
$985$	−5.23281e8			−0.547553
$986$	−8.95927e8	−	1.74189e9i	−0.934635	−	1.81714i
$987$		0			0
$988$	−9.76429e7	−	6.98092e7i	−0.101244	−	0.0723839i
$989$			1.89078e9i			1.95457i
$990$		0			0
$991$			3.88620e8i			0.399304i	0.979867	+	0.199652i	$0.0639812\pi$
$991$			3.88620e8i			0.399304i	−0.979867	+	0.199652i	$0.936019\pi$
$992$	1.00236e9	−	1.04136e9i	1.02681	−	1.06676i
$993$		0			0
$994$	1.05882e9	−	5.44597e8i	1.07811	−	0.554519i
$995$	3.15772e8			0.320556
$996$		0			0
$997$			1.30107e9i			1.31285i	0.754390	+	0.656426i	$0.227933\pi$
$997$			1.30107e9i			1.31285i	−0.754390	+	0.656426i	$0.772067\pi$
$998$	3.22971e7	−	1.66118e7i	0.0324917	−	0.0167118i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	72.7.b.c.19.4		12
3.2	odd	2		24.7.b.a.19.9	✓	12
4.3	odd	2		288.7.b.d.271.8		12
8.3	odd	2	inner	72.7.b.c.19.3		12
8.5	even	2		288.7.b.d.271.5		12
12.11	even	2		96.7.b.a.79.9		12
24.5	odd	2		96.7.b.a.79.10		12
24.11	even	2		24.7.b.a.19.10	yes	12
48.5	odd	4		768.7.g.l.511.16		24
48.11	even	4		768.7.g.l.511.14		24
48.29	odd	4		768.7.g.l.511.13		24
48.35	even	4		768.7.g.l.511.15		24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
24.7.b.a.19.9	✓	12	3.2	odd	2
24.7.b.a.19.10	yes	12	24.11	even	2
72.7.b.c.19.3		12	8.3	odd	2	inner
72.7.b.c.19.4		12	1.1	even	1	trivial
96.7.b.a.79.9		12	12.11	even	2
96.7.b.a.79.10		12	24.5	odd	2
288.7.b.d.271.5		12	8.5	even	2
288.7.b.d.271.8		12	4.3	odd	2
768.7.g.l.511.13		24	48.29	odd	4
768.7.g.l.511.14		24	48.11	even	4
768.7.g.l.511.15		24	48.35	even	4
768.7.g.l.511.16		24	48.5	odd	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 72 = 2^{3} \cdot 3^{2} \)
Weight:	\( k \)	\(=\)	\( 7 \)
Character orbit:	\([\chi]\)	\(=\)	72.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+(-7.11414 + 3.65910i) q^{2} +(37.2219 - 52.0627i) q^{4} +87.0704i q^{5} -355.505i q^{7} +(-74.2991 + 506.580i) q^{8} +O(q^{10})\)	\(q+(-7.11414 + 3.65910i) q^{2} +(37.2219 - 52.0627i) q^{4} +87.0704i q^{5} -355.505i q^{7} +(-74.2991 + 506.580i) q^{8} +(-318.599 - 619.431i) q^{10} +185.285 q^{11} +2323.93i q^{13} +(1300.83 + 2529.11i) q^{14} +(-1325.06 - 3875.75i) q^{16} -7535.06 q^{17} -807.031 q^{19} +(4533.12 + 3240.93i) q^{20} +(-1318.14 + 677.978i) q^{22} -15162.9i q^{23} +8043.75 q^{25} +(-8503.52 - 16532.8i) q^{26} +(-18508.6 - 13232.6i) q^{28} -32494.6i q^{29} -44109.9i q^{31} +(23608.4 + 22724.1i) q^{32} +(53605.5 - 27571.6i) q^{34} +30954.0 q^{35} -59093.4i q^{37} +(5741.33 - 2953.01i) q^{38} +(-44108.1 - 6469.25i) q^{40} -87145.9 q^{41} -124698. q^{43} +(6896.68 - 9646.45i) q^{44} +(55482.5 + 107871. i) q^{46} +64962.9i q^{47} -8735.01 q^{49} +(-57224.4 + 29432.9i) q^{50} +(120990. + 86501.3i) q^{52} -154374. i q^{53} +16132.9i q^{55} +(180092. + 26413.7i) q^{56} +(118901. + 231171. i) q^{58} -41543.6 q^{59} +108362. i q^{61} +(161403. + 313804. i) q^{62} +(-251103. - 75277.0i) q^{64} -202346. q^{65} +482658. q^{67} +(-280469. + 392296. i) q^{68} +(-220211. + 113264. i) q^{70} -418654. i q^{71} -546155. q^{73} +(216229. + 420399. i) q^{74} +(-30039.2 + 42016.2i) q^{76} -65869.9i q^{77} +538230. i q^{79} +(337463. - 115373. i) q^{80} +(619968. - 318876. i) q^{82} -443538. q^{83} -656080. i q^{85} +(887119. - 456283. i) q^{86} +(-13766.5 + 93861.9i) q^{88} +151438. q^{89} +826171. q^{91} +(-789420. - 564391. i) q^{92} +(-237706. - 462155. i) q^{94} -70268.4i q^{95} -1.18214e6 q^{97} +(62142.0 - 31962.3i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 10 q^{2} + 24 q^{4} - 796 q^{8}+O(q^{10}) \) 12 * q - 10 * q^2 + 24 * q^4 - 796 * q^8	\( 12 q - 10 q^{2} + 24 q^{4} - 796 q^{8} + 2172 q^{10} - 2720 q^{11} + 6444 q^{14} + 11640 q^{16} + 4888 q^{17} + 3936 q^{19} + 31608 q^{20} - 60432 q^{22} - 27204 q^{25} - 53952 q^{26} - 57072 q^{28} - 109480 q^{32} + 47388 q^{34} - 162336 q^{35} + 89080 q^{38} + 72120 q^{40} + 54280 q^{41} - 49824 q^{43} - 229184 q^{44} + 171864 q^{46} - 304644 q^{49} + 500078 q^{50} + 256848 q^{52} + 699816 q^{56} - 409524 q^{58} + 886144 q^{59} - 691356 q^{62} - 500640 q^{64} - 473376 q^{65} + 1565952 q^{67} - 669104 q^{68} + 473784 q^{70} + 555480 q^{73} + 753720 q^{74} - 293136 q^{76} + 251616 q^{80} + 2317716 q^{82} - 2497760 q^{83} - 476024 q^{86} + 971424 q^{88} - 367400 q^{89} - 4475808 q^{91} + 377376 q^{92} - 2642568 q^{94} - 1165656 q^{97} - 182674 q^{98}+O(q^{100}) \) 12 * q - 10 * q^2 + 24 * q^4 - 796 * q^8 + 2172 * q^10 - 2720 * q^11 + 6444 * q^14 + 11640 * q^16 + 4888 * q^17 + 3936 * q^19 + 31608 * q^20 - 60432 * q^22 - 27204 * q^25 - 53952 * q^26 - 57072 * q^28 - 109480 * q^32 + 47388 * q^34 - 162336 * q^35 + 89080 * q^38 + 72120 * q^40 + 54280 * q^41 - 49824 * q^43 - 229184 * q^44 + 171864 * q^46 - 304644 * q^49 + 500078 * q^50 + 256848 * q^52 + 699816 * q^56 - 409524 * q^58 + 886144 * q^59 - 691356 * q^62 - 500640 * q^64 - 473376 * q^65 + 1565952 * q^67 - 669104 * q^68 + 473784 * q^70 + 555480 * q^73 + 753720 * q^74 - 293136 * q^76 + 251616 * q^80 + 2317716 * q^82 - 2497760 * q^83 - 476024 * q^86 + 971424 * q^88 - 367400 * q^89 - 4475808 * q^91 + 377376 * q^92 - 2642568 * q^94 - 1165656 * q^97 - 182674 * q^98

Introduction

L-functions

Modular forms

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Database

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